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Properties

Label 2-637-91.19-c1-0-17
Degree $2$
Conductor $637$
Sign $0.913 + 0.406i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 − 0.638i)2-s − 0.168i·3-s + (3.53 + 2.04i)4-s + (2.38 − 0.638i)5-s + (−0.107 + 0.402i)6-s + (−3.63 − 3.63i)8-s + 2.97·9-s − 6.08·10-s + (3.22 + 3.22i)11-s + (0.344 − 0.596i)12-s + (1.54 − 3.25i)13-s + (−0.107 − 0.402i)15-s + (2.24 + 3.89i)16-s + (−0.0563 + 0.0976i)17-s + (−7.07 − 1.89i)18-s + (2.43 + 2.43i)19-s + ⋯
L(s)  = 1  + (−1.68 − 0.451i)2-s − 0.0974i·3-s + (1.76 + 1.02i)4-s + (1.06 − 0.285i)5-s + (−0.0440 + 0.164i)6-s + (−1.28 − 1.28i)8-s + 0.990·9-s − 1.92·10-s + (0.971 + 0.971i)11-s + (0.0994 − 0.172i)12-s + (0.429 − 0.903i)13-s + (−0.0278 − 0.103i)15-s + (0.562 + 0.973i)16-s + (−0.0136 + 0.0236i)17-s + (−1.66 − 0.447i)18-s + (0.558 + 0.558i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.913 + 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931004 - 0.198000i\)
\(L(\frac12)\) \(\approx\) \(0.931004 - 0.198000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-1.54 + 3.25i)T \)
good2 \( 1 + (2.38 + 0.638i)T + (1.73 + i)T^{2} \)
3 \( 1 + 0.168iT - 3T^{2} \)
5 \( 1 + (-2.38 + 0.638i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.22 - 3.22i)T + 11iT^{2} \)
17 \( 1 + (0.0563 - 0.0976i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 2.43i)T + 19iT^{2} \)
23 \( 1 + (0.565 - 0.326i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.82 - 4.88i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.43 - 5.34i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.402 + 1.50i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (10.6 - 2.86i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.08 + 3.51i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.53 - 5.72i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.41 + 4.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.02 - 3.81i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 15.3iT - 61T^{2} \)
67 \( 1 + (-4.44 + 4.44i)T - 67iT^{2} \)
71 \( 1 + (-3.56 - 0.955i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.43 - 0.651i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.11 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.34 + 3.34i)T + 83iT^{2} \)
89 \( 1 + (-8.39 - 2.24i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.04 + 3.89i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23403142829201700441294189977, −9.611538556296461289846604371678, −9.141490175968625474605493771643, −8.073141822913603371619729826016, −7.21335241550008294701042559349, −6.43706618398294272016424644050, −5.12713106971005366125281929726, −3.50743868201365130971362096975, −1.90062076774101457459764777363, −1.30698879891129126786970818557, 1.14765114647407644011481989129, 2.18443283250051470238453288077, 3.98392880715453591562605384240, 5.71225122433129998757374156907, 6.49602882583494971304366650878, 7.08196979310801955520116259158, 8.187853979702083481512668899395, 9.180899759070089485552116313802, 9.513256775190280165952924502537, 10.28441645870758841096184789635

Graph of the $Z$-function along the critical line